Problem: Let $y=\tan(x)$. What is the value of $\dfrac{dy}{dx}$ at $x=\dfrac{2\pi}{3}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $4$ (Choice B) B $\dfrac14$ (Choice C) C $-4$ (Choice D) D $-\dfrac{1}{4}$
Let's first find $\dfrac{dy}{dx}$. Then, we can evaluate it at $x=\dfrac{2\pi}{3}$. Recall that the derivative of $\tan(x)$ is $\dfrac{1}{\cos^2(x)}$, or $\sec^2(x)$. Put another way, $\dfrac{d}{dx}[\tan(x)]=\dfrac{1}{\cos^2(x)}=\sec^2(x)$. [Is there a way to know this without memorizing?] Now let's plug in $x={\dfrac{2\pi}{3}}$ : $\begin{aligned} &\phantom{=}\dfrac{1}{\cos^2\left({\dfrac{2\pi}{3}}\right)} \\\\ &=\dfrac{1}{\left(-\dfrac{1}{2}\right)^2} \\\\ &=\dfrac{1}{\left(\dfrac{1}{4}\right)} \\\\ &=4 \end{aligned}$ In conclusion, the value of $\dfrac{dy}{dx}$ at $x=\dfrac{2\pi}{3}$ is $4$.